In our filters,
we use a variational approximation of the KR rearrangement to Gaussianize
a collection of particles.
The idea of Gaussianizing multivariate data is
well rooted in
statistics [1]}, and
is known as Gaussian anamorphosis (GA)
in the geostatistics literature [2]}.
Usually, these approaches Gaussianize marginals of the data
by estimating one-dimensional cumulative distribution functions.
A typical application of GA within the context of non-Gaussian extensions of the
EnKF is described in [3]}:
the idea is to (1) estimate a nonlinear transformation that
Gaussianizes joint samples from the distribution of state and data
at a given observation time, (2) derive an equivalent nonlinear
observation operator in the transformed space, (3) apply EnKF formulas
for nonlinear observation operators in the transformed space, and (4) map the analysis ensemble
back to the original space.
As explained in [3]}, this approach cannot
recover the exact filtering distribution for non-Gaussian models, even if we
had the exact nonlinear transformation.
The problem is that we insist on using EnKF formulas in the transformed space.
In the stochastic map filter introduced in this paper, on the other hand, we show how to bypass this
issue by designing a nonlinear update that depends only on the
nonlinear transformation that Gaussianizes the data.
If we had the exact transformation, then the stochastic map
filter would be exact.
